Chaotic dynamics in an infinite-dimensional electromagnetic system

AbstractThe understanding of chaotic dynamics in high-dimensional systems that has emerged in the last decade offers a promising dynamical framework to study turbulence. Here turbulence is viewed as a walk through a forest of exact solutions in the infinite-dimensional state space of the governing equations. Recently, Chandler & Kerswell (J. Fluid Mech., vol. 722, 2013, pp. 554–595) carry out the most exhaustive study of this programme undertaken so far in fluid dynamics, a feat that requires every tool in the dynamicist’s toolbox: numerical searches for recurrent flows, computation of their stability, their symmetry classification, and estimating from these solutions statistical averages over the turbulent flow. In the long run this research promises to develop a quantitative, predictive description of moderate-Reynolds-number turbulence, and to use this description to control flows and explain their statistics.

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Complexity in Linear Systems: A Chaotic Linear Operator on the Space of Odd2π-Periodic Functions

Complexity ◽

10.1155/2017/6020213 ◽

2017 ◽

Vol 2017 ◽

pp. 1-8 ◽

Cited By ~ 2

Author(s):

Tamás Kalmár-Nagy ◽

Márton Kiss

Keyword(s):

Nonlinear Systems ◽

Linear Operator ◽

Linear Systems ◽

Chaotic Dynamics ◽

Periodic Points ◽

Complex Behavior ◽

Periodic Functions ◽

Infinite Dimensional ◽

Backward Shift ◽

Principal Value

Not just nonlinear systems but infinite-dimensional linear systems can exhibit complex behavior. It has long been known that twice the backward shift on the space of square-summable sequencesl2displays chaotic dynamics. Here we construct the corresponding operatorCon the space of2π-periodic odd functions and provide its representation involving a Principal Value Integral. We explicitly calculate the eigenfunction of this operator, as well as its periodic points. We also provide examples of chaotic and unbounded trajectories ofC.

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Fixed Points and Chaotic Dynamics of an Infinite Dimensional Map

Chaos, Noise and Fractals ◽

10.1201/9781003069553-10 ◽

2020 ◽

pp. 137-186

Author(s):

J V Moloney ◽

H Adachihara ◽

D W McLaughlin ◽

A C Newell

Keyword(s):

Fixed Points ◽

Chaotic Dynamics ◽

Infinite Dimensional

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Chaotic dynamics for perturbations of infinite-dimensional Hamiltonian systems

Nonlinear Analysis ◽

10.1016/s0362-546x(00)00200-5 ◽

2002 ◽

Vol 48 (4) ◽

pp. 481-504 ◽

Cited By ~ 6

Author(s):

Massimiliano Berti ◽

Carlo Carminati

Keyword(s):

Hamiltonian Systems ◽

Chaotic Dynamics ◽

Infinite Dimensional

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Relative periodic orbits form the backbone of turbulent pipe flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2017.699 ◽

2017 ◽

Vol 833 ◽

pp. 274-301 ◽

Cited By ~ 23

Author(s):

N. B. Budanur ◽

K. Y. Short ◽

M. Farazmand ◽

A. P. Willis ◽

P. Cvitanović

Keyword(s):

Periodic Orbits ◽

Pipe Flow ◽

Chaotic Dynamics ◽

Stokes Equations ◽

Turbulent Pipe Flow ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Infinite Dimensional ◽

Unstable Manifolds ◽

Low Dimensional

The chaotic dynamics of low-dimensional systems, such as Lorenz or Rössler flows, is guided by the infinity of periodic orbits embedded in their strange attractors. Whether this is also the case for the infinite-dimensional dynamics of Navier–Stokes equations has long been speculated, and is a topic of ongoing study. Periodic and relative periodic solutions have been shown to be involved in transitions to turbulence. Their relevance to turbulent dynamics – specifically, whether periodic orbits play the same role in high-dimensional nonlinear systems like the Navier–Stokes equations as they do in lower-dimensional systems – is the focus of the present investigation. We perform here a detailed study of pipe flow relative periodic orbits with energies and mean dissipations close to turbulent values. We outline several approaches to reduction of the translational symmetry of the system. We study pipe flow in a minimal computational cell at $Re=2500$, and report a library of invariant solutions found with the aid of the method of slices. Detailed study of the unstable manifolds of a sample of these solutions is consistent with the picture that relative periodic orbits are embedded in the chaotic saddle and that they guide the turbulent dynamics.

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